Uchicago Geometry 1 Ps-04

8/10/2019 Uchicago Geometry 1 Ps-04

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Algebraic topology (Autumn 2014)

The class hours are MWF 9-10:20 in Eck 206.Homework is due Monday morning.

There will be a midterm (Wednesday October 29) and a final.

I will hold office hours at 10:30, and occasional other times -- also by appointment in myoffice , Eck 403. My email is [email protected]

Bena Tshishiku ([email protected]) and Nick Salter ([email protected]) will

be doing the grading and giving office hours (alternate weeks). They will have officehours on Friday 11-12. Bena will do odd weeks and Nick even. Benas will be in Jones

205 and Nicks in Eckhart 11.

Recommended texts are by May, Hatcher and Spanier.

Monday October 20 there will be no class. Also Wednesdays November 12 and 19.

Homework 1. (Do 7 or as much as you can)

1. Show that any subgroup G of a free group F is free. Show that the rank, if theindex is finite, only depends on [F: G]. Give a counterexample for infinite index.

2. Suppose f: S1!S

1is a continuous map, so that f(f(x)) = x, and f has no fixed

points, show that there is a homeomorphism h: S1!S1so that hfh-1(x) = -x. (for all

x)

3.

Suppose that f: C!

C is of the form f(z) = z

n

+ h(z) and lim h(z)/||z||

n

= 0, showthat f has a root (i.e. a solution to the equation f(x) = 0). Can it have infinitelymany roots?

4. What is the universal cover of RP2!RP

2? Write down the covering map. What

is !2(RP2!RP

2)?

5. Suppose X is simply connected and f:X !C is a nowhere 0 function. Show that

log(f) can be defined to be a continuous function. Is there a nonsimply connectedX for which this is true?

6.

Suppose f: M !N is a smooth map between compact smooth manifolds of thesame dimension, and Jac(f) is nowhere vanishing, M connected and N is simplyconnected, show that f is a diffeomorphism. Show that if M is not compact, this

need not be true.

7. Prove that the (p,q) torus knot is not isomorphic to the (r,s) torus knot unless {p,q} = {r, s}. Recall that the (a,b) torus knot is the embedding of S

1in S

3which


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lies in the standard unknotted torus, via z !(za,z

b). (Here a and b are relatively

prime integers > 1.)

8. Let PU(n) = SU(n)/Z where Z is the center of SU(n) (the group of unitary n x n

complex matrices with determinant = 1). What is !1(PU(n))?

9. Consider S1!S

2!D

2!S

2 What is an element of !2that is nontrivial in the

domain, but becomes trivial in the range? (There is an old conjecture that for 2

complexes X!Y, if !2Y = 0, then so is !2X. This exercise shows that the

conjecture cannot be that inclusions of 2-complexes give injections of !2.)

10.Suppose that X is a contractible polyhedron, show that for all k there is a functionf:X

k!X that commutes with permuting the coordinates, and has f(x,...x) = x.

Show that for X = RP2no such f exists for k=2.

Homework 2 . (Do 7 or as much as you can)

1. Give maps from S1into the figure 8 that are homotopic but not pointed

homotopic.

2. Suppose X and Y are simply connected and homotopy equivalent, then they are

pointed homotopy equivalent. (Assume X and Y are decent according to yourown reasonable definition of decency). (*) Is this true without simple

connectivity?

3. Let X be decent and connected. Consider the set of homotopy classes of mapsfrom X to S

1which send a base point to 1. Call this !

1(X). Show that

a. !1(X) is an abelian group using pointwise multiplication on the circle.

b. If f: X !Y is a map, then there is an iduced map f*: !1(Y) !!

1(X). (fg)*

= g*f*.

c. !1(X) is determined by the fundamental groups of the components of X.

Whats a formula?d. Show that there is a homotopy equivalence between Maps(X: S

1) and

!1(X).

4. Definition: A line bundle over X is a pair (Z,f), where f: Z !X is a map so that X

has a cover by open sets Oi, so that over Oi one has f-1

(Oi) !Oi"R so that with

this identification f is projection to the first factor. One also assumes that overOi"Oj the map Oi"Oj"R !Oj"Oi"R is linear over each point. Two line

bundles are isomorphic if there is a map h: Z!Z so that f = fh and h is linear on

each fiber (inverse image of a point of X).

Show that there is an invariant of line bundles taking values in Hom(!1X : Z/2Z)obtained by sending a line bundle to the following (not quite) covering space. Send Zto Z -0-section and mod out by the multiplicative action of positive real numbers.


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(Can you show that this invariant is complete?)

5. Suppose Z is a compact subset of Euclidean space and a neighborhood retract.

Show that there is an #>0, so that for any X, if f,g:X!Z have d(f,g) < #, then f and

g are homtopic and that the same is true rel A for any A in X. Deduce that the setof homotopy classes of maps from X to Z is always countable (for X a compactmetric space).

6. (continuation) Same Z as in #5. Show that for any L, there are only finitely many

homotopy classes of maps in !d(Z) with Lipschitz constant at most L. Indeed,show that the number grows at most like exp(L

d).

7. A space X is an H-space if there is a continuous function :X $X %X, for which

for a suitable e"X, (e,x) = (x,e) = x. Show that !1(X,e) is abelian if X is an H-space.

8. Suppose G and H are groups, show that any finite subgroup of G*H is conjugate

to a subgroup of either G or of H.

Remark: A useful lemma is that no nontrivial free product is finite. If g is in G and h

is in H, (both nontrivial), then gh has infinite order.

9. Show that S3$CP

2and S

2$S

5have isomorphic homotopy groups. (These

manifolds are nothomotopy equivalent. Why does this not contradict the

Whitehead theorem?)

Hint: You might first think about the analogous problem for S

3

$RP

2

and S

2

$RP

3

.

10.If M and N are connected n-manifolds, the result of removing little open n-ballsfrom M and N and then glueing them together is called the connected sum of M

and N and denoted by M#N. (Actually there are two isotopy classes of glueingmaps, so if M and N are orientable, one must use the orientations to make a well

defined construction.) Show that if n>2, !1(M#N) = !1M *!1N. Why is this not

true for n=2? Give an example of a connected sum of 2-manifolds that is not a

free product.

Remark: If n=3 a wonderful theorem of Stallings1asserts the converse of the first

part of this problem: A closed 3 dimensional manifold is a connected sum iff itsfundamental group is a nontrivial free product. Similarly, Stallings showed that amap f:M

3%S

1is homotopic to the projection of a fiber bundle iff f* is nontrivial and

its kernel is finitely generated. Do you see why this is necessary?

1In combination with the Poincare conjecture. (With PC, the statement is slightly more

complicated.)


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11.Show that no infinite group acts properly discontinuously on the Moebius strip.(Hint: Show that any homeomorphism h of the Moebius strip satisfies the

condition h(C) "C .)

Homework 3 (Do 5 problems)

1. Suppose X is a 2-complex and !2X = 0, then show that !3X = 0.

2. Let Lk2n-1

(a1....an) and Lk2n-1

(b1....bn) be two lens spaces (of the same dimension

and same fundamental group). Show that there is a map f: Lk2n-1

(a1....an)%-Lk

2n-

1(b1....bn) that induces an isomorphism on !1.

3. If A, B and C are groups and i:C %A and j: C %B are injections induced by

maps of spaces K(C,1) %K(A,1) and K(C,1) %K(B,1), then show that K(A,1)

!K(C,1)K(B,1) is a K(A*CB, 1). Give a counterexample if they are not injections.

4. Show that any complex line bundle over Snis trivial for n>2. (Change the

definition from 2.4 from an Rto a C.)

5. Show that if f:S1%S

1satisfies f(x) = -f(-x), then the element of !1(S

1) "Z is

odd. If f(x) = f(-x) then it is even. (These are both true for Snfor all n; you will

be able to prove this in a few weeks).

6. Show that '(T2) is homotopy equivalent to S

2!S

2!S

3(Hint: use a cell

decomposition; remember !2is abelian.)

7.

Show that k and l are relatively prime iff every map from Lk2n-1(1,1,...1) toLl

2n+1(1,....1) is null homotopic.

8. What is !2(RP2, RP

1)?

9. Let SO(n) be the group of orthogonal matrices of determinant 1. Check that

SO(2) is the circle. Observe that SO(n) acts on Sn-1

transitively (i.e. with a singleorbit) and with isotropy SO(n-1). Deduce SO(n) is connected for all n. Show that

SO(n) %SO(n+1) induces an isomorphism on !ifor all i


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1. Show that any map T2%8 (The figure 8) is homotopic to one that is not

surjective. Show that the identity map 8%8 is not homotopic to a map that is

not surjective.

2. Let X,x be a pointed space and let )X = the space of maps f:[0,1] %X, such that

f(0) = f(1) = x. Show that !i()X)!

!i+1()X). More generally, for any pointed Z,the homotopy classes of pointed maps ['Z %X] is isomorphic to the homotopy

classes of pointed maps [Z %)X]

3. Show that if X is a CW complex with !i(X) = 0 for ik, then X is homotopy

equivalent to a CW complex Z, with Zk= a point.

For the remaining problems you can assume that you know that !n(Sn) = Z.

4. Show that for every n there is a space K(Z, n) with the property that !i(K(Z, n)) =

0 unless i=n and !n(K(Z, n)) = Z. Show that this space is uniquely determined up

to homotopy type by n.

5. Show that for all n, the space K(Z, n) has a multiplication :K$K %K with anidentity, i.e. (k, e) = (e, k) = k.

6. Show that )K(Z, n) = K(Z, n-1); using this show that for any X, the pointed maps

[X: K(Z, n)] forms an abelian group. We shall denote this group En(X).

7. Show that if X is a k-dimensional CW complex, then En(X) = 0 for n>k.

8. Show that Ek+1

('X) !Ek(X)

9. Show that if A!X is an inclusion of CW complexes, then there is an exact

sequence ...%Ek(X/A) %E

k(X) %E

k(A) %E

k+1(X/A) %...

The map Ek(A) %E

k+1(X/A) is induced by considering X/A as X#AcA and using any

map X %cA that is the identity on A (why is there such a thing) to give a map

X/A%'A, and making use of problem 8.

10.Show that there is an exact sequence if Z = X #Y and A = X "Y, all CW

complexes:

%Ek(Z) %E

k(X)E

k(Y) %E

k(A) %E

k+1(Z) %...

11.Compute all Ek(RP

n) (by induction on n). Deduce that there is no finite

dimensional K(Z/2Z, 1).

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Uchicago Geometry 1 Ps-04